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Some properties of meromorphically multivalent functions
Journal of Inequalities and Applications volume 2012, Article number: 86 (2012)
Abstract
By using the method of differential subordinations, we derive certain properties of meromorphically multivalent functions.
2010 Mathematics Subject Classification: 30C45; 30C55.
1 Introduction
Let Σ(p) denotes the class of meromorphically multivalent functions f(z) of the form
which are analytic in the punctured unit disk
Let f(z) and g(z) be analytic in U. Then, we say that f(z) is subordinate to g(z) in U, written f(z) ≺ g(z), if there exists an analytic function w(z) in U, such that |w(z)| ≤ |z| and f(z) = g(w(z)) (z ∈ U). If g(z) is univalent in U, then the subordination f(z) ≺ g(z) is equivalent to f(0) = g(0) and f(U) ⊂ g(U).
Let p(z) = 1 + p1z + ... be analytic in U. Then for -1 ≤ B < A ≤ 1, it is clear that
if and only if
and
Recently, several authors (see, e.g., [1–7]) considered some interesting properties of meromorphically multivalent functions. In the present article, we aim at proving some subordination properties for the class Σ(p).
To derive our results, we need the following lemmas.
Lemma 1 (see [8]. Let h(z) be analytic and starlike univalent in U with h(0) = 0. If g(z) is analytic in U and zg'(z) ≺ h(z), then
Lemma 2 (see [9]. Let p(z) be analytic and nonconstant in U with p(0) = 1. If 0 < | z0 | < 1 and , then
2 Main results
Our first result is contained in the following.
Theorem 1. Let and β ∈ (0,1). If f(z) ∈ Σ(p) satisfies f(z) ≠0 (z ∈ U*) and
where δ is the minimum positive root of the equation
then
The bound β is the best possible for each .
Proof. Let
We can see that the Equation (2.2) has two positive roots. Since g(0) > 0 and g(1) < 0, we have
Put
Then from the assumption of the theorem, we see that p(z) is analytic in U with p(0) = 1 and α + (1 - α)p(z) ≠0 for all z ∈ U. Taking the logarithmic differentiations in both sides of (2.6), we get
and
for all z ∈ U. Thus the inequality (2.1) is equivalent to
By using Lemma 1, (2.9) leads to
or to
In view of (2.5), (2.10) can be written as
Now by taking and B = -δ in (1.2) and (1.3), we have
for all z ∈ U because of g(δ) = 0. This proves (2.3).
Next, we consider the function f(z) defined by
It is easy to see that
Since
it follows from (1.3) that
Hence, we conclude that the bound β is the best possible for each .
Next, we derive the following.
Theorem 2. If f(z) ∈ Σ(p) satisfies f(z) ≠0 (z ∈ U*) and
where
then
The bound in (2.14) is sharp.
Proof. Let
Then p(z) is analytic in U with p(0) = 1 and p(z) ≠0 for z ∈ U. In view of (2.15) and (2.12), we have
i.e.,
Now by using Lemma 1, we obtain
Since the function 1 - 2γ log(1 - z) is convex univalent in U and
from (2.16), we get the inequality (2.14).
To show that the bound in (2.14) cannot be increased, we consider
It is easy to verify that the function f(z) satisfies the inequality (2.12). On the other hand, we have
as z → -1. Now the proof of the theorem is complete.
Finally, we discuss the following theorem.
Theorem 3. Let f(z) ∈ Σ(p) with f(z) ≠0 (z ∈ U*). If
for some λ(λ > 0), then
Proof. Let us define the analytic function p(z) in U by
Then p(0) = 1, p(z) ≠0 (z ∈ U) and
Suppose that there exists a point z0(0 < | z0 | < 1) such that
where β is real and β ≠0. Then, applying Lemma 2, we get
Thus it follows from (2.19), (2.20), and (2.21) that
In view of λ > 0, from (2.21) and (2.22) we obtain
and
But both (2.23) and (2.24) contradict the assumption (2.17). Therefore, we have Rep(z) > 0 for all z ∈ U. This shows that (2.18) holds true.
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Xu, YH., Yang, Q. & Liu, JL. Some properties of meromorphically multivalent functions. J Inequal Appl 2012, 86 (2012). https://doi.org/10.1186/1029-242X-2012-86
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DOI: https://doi.org/10.1186/1029-242X-2012-86